The Stacks project

Lemma 27.20.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent:

  1. $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module of finite presentation,

  2. $\mathcal{F}$ is $\mathcal{O}_ X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module,

  3. $\mathcal{F}$ is a locally free, finite type $\mathcal{O}_ X$-module,

  4. $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module, and

  5. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type, for every $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module, and the function

    \[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

    is locally constant in the Zariski topology on $X$.

Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 10.77.2. The translation uses Lemmas 27.16.1, 27.16.2, 27.19.1, and 27.20.1. $\square$


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